Abstract
Modeling ferroelectric materials from first principles is one of the successes of densityfunctional theory and the driver of much development effort, requiring an accurate description of the electronic processes and the thermodynamic equilibrium that drive the spontaneous symmetry breaking and the emergence of macroscopic polarization. We demonstrate the development and application of an integrated machine learning model that describes on the same footing structural, energetic, and functional properties of barium titanate (BaTiO_{3}), a prototypical ferroelectric. The model uses ab initio calculations as a reference and achieves accurate yet inexpensive predictions of energy and polarization on time and length scales that are not accessible to direct ab initio modeling. These predictions allow us to assess the microscopic mechanism of the ferroelectric transition. The presence of an orderdisorder transition for the Ti offcentered states is the main driver of the ferroelectric transition, even though the coupling between symmetry breaking and cell distortions determines the presence of intermediate, partlyordered phases. Moreover, we thoroughly probe the static and dynamical behavior of BaTiO_{3} across its phase diagram without the need to introduce a coarsegrained description of the ferroelectric transition. Finally, we apply the polarization model to calculate the dielectric response properties of the material in a full ab initio manner, again reproducing the correct qualitative experimental behavior.
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Introduction
Ferroelectric materials possess a spontaneous electric polarization that can be switched with an external electric field. The discovery of ferroelectricity in barium titanate (BaTiO_{3}), the prototypical ferroelectric perovskite, changed the general understanding and perception of ferroelectrics due in large part to its relatively simple crystal structure^{1}. At low temperatures, BaTiO_{3} is rhombohedral with polarization along the 〈111〉 direction; at higher temperatures, it undergoes three phase transitions, first to an orthorhombic phase with the polarization along the 〈110〉 direction at 183 K, then to a tetragonal phase with the polarization along 〈100〉 at 278 K, and finally, at 393 K, to a cubic, paraelectric phase^{2}. It has been long understood that the spontaneous polarization is a result of the titanium atom offcentering within the enclosing oxygen octahedron, but the detailed microscopic nature of the ferroelectric transition has been the subject of intense, ongoing research with a variety of experimental and theoretical techniques. The ferroelectric transitions were first described with a displacive model in which the Tidisplacements are driven by a transverse phonon instability^{3}. Almost concurrently, an orderdisorder model was proposed to explain the origin of the Tidisplacements along any one of the eight local 〈111〉 directions in the cubic phase, as driven by the pseudo JahnTeller effect ^{4}, showing how these displacements order at lower temperatures in different ferroelectric phases^{5,6}. These models capture some of the phenomena experimentally observed in characterizing BaTiO_{3}, such as phonon softening at the transition temperatures^{7,8} —consistent with the displacive model—and diffuse Xray scattering in all phases except the rhombohedral one^{9,10,11}—consistent with the orderdisorder model—leading also to approaches combining the two models^{12,13,14}. In this context, simulations – especially from first principles—can offer a precious microscopic understanding of the nature of the phase transitions.
A computer simulation of the ferroelectric phase transition of any given material requires three key ingredients: first, a model of the potential energy surface (PES) that describes the energetic response to atomic and structural distortions, second, the freeenergy surface (FES) sampled at the relevant, finitetemperature thermodynamic conditions, and third, the polarization of individual configurations that determines, through averaging over samples, the macroscopic polarization.
Densityfunctional theory (DFT) calculations have long been used to explore the PES of BaTiO_{3} as well as the soft phonons and their strong dependence on pressure^{15,16,17,18}. Further DFT investigations have found that Tidisplacements along local 〈111〉 directions can result in dynamically stable structures^{19,20,21}. The phase transitions and rhombohedralorthorhombictetragonalcubic (ROTC) phase sequence of BaTiO_{3} has been extensively studied and reproduced using effective Hamiltonians solved using both MonteCarlo^{22,23} and molecular dynamics (MD)^{24,25,26}; furthermore, similar studies have been carried out on other perovskite systems^{27}, including solid solutions^{28}. Despite their successes, effective models rely on the choice of an explicit parametrization of the Hamiltonian; therefore, in order to confidently make firstprinciplesaccurate predictions of the thermodynamics, it is desirable to use an unbiased, agnostic approach without any prior assumption in the form of the PES.
To this aim, we introduce an integrated machine learning (ML) framework allowing us to carry out MD without the need to compromise on simulation size and time scales. This framework, based on a combination of an interatomic ML potential and a vector ML model for the polarization, is used to simultaneously predict the total energy, atomic forces, and polarization of a ferroelectric material in order to explore its complex, temperaturedependent phase diagram as well as to predict its functional properties. This approach allows us to compute macroscopic observables—chemical potentials and dielectric susceptibilities, specifically—with an accuracy equivalent to that of the level of theory of the underlying DFT calculations, but at a much smaller computational cost. Moreover, it is applicable with only minor changes to any perovskite or even any other type of ferroelectric material, including 2D ferroelectrics^{29}. Although we do not reach a quantitative agreement with the experimental ROTC transition temperatures, we demonstrate that this limitation in accuracy stems from the DFT reference itself and not the approximation made in modeling the potential energy surface. Thus, we foresee clear, systematic pathways to improving the model potential, with only slight modifications of the ML methodology. Specifically, the generality of the framework and the relatively small size of the training dataset make it possible to improve the model accuracy by computing the reference structures with more advanced functionals such as Hubbardcorrected DFT^{30,31}, metaGGAs,^{32} and hybrids^{33}.
The key advance underlying this work is a unified ML framework combining an interatomic potential based on the SOAPGAP method^{34} and a microscopic polarization model based on the symmetryadapted Gaussian process regression (SAGPR) method^{35}. The use of ML for materials modeling has gained considerable momentum in the past decade^{34,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56}. Specifically, the prediction of finitetemperature properties of materials as the ones we focus on in this paper relies on the construction of ML potential energy surfaces based on a set of reference structures computed with ab initio methods^{53,57,58}. Such potentials allow the simulation of molecules and complex solids with almost the same accuracy as the reference method used to generate the dataset. In this way, it is possible to investigate the meso and macroscopic properties of materials at a considerably reduced computational effort compared to direct ab initio simulations. Notable successes of the machine learning potentials approach include the study of bulk and interfacial properties of metallic alloys from cryogenic temperatures up to the melting point^{45}; finitetemperature modeling of binary systems with variable concentration, such as GaAs^{46}; accurate calculations on the relative stability of competing phases of various compounds, such as sodium^{58}, carbon^{59}, water^{47}, iron^{49}, and silicon^{50}; as well as MD studies of polycrystalline phasechange materials^{57} and hybrid perovskites^{60}.
Two important developments have enabled the great success of machine learning in condensed matter and chemical physics. First, appropriate regression schemes—such as kernel methods, typified by Gaussian approximation potentials (GAP)^{34}; neural networks (e.g., of the Behler–Parrinello type^{36} or more recent graph convolutional approaches^{43,61}); or nonkernelbased linear fitting schemes (with appropriate representations^{62,63,64,65})—have been designed and specialized for atomistic systems. The key to nearly all of these methods is the decomposition of a global (extensive) physical observable of the system into local contributions, each written as a function of the neighborhood of individual atoms. Note that this decomposition carries with it an implicit assumption of the locality of the potential energy surface, thus neglecting the effect of longrange electrostatic and dispersion forces. Several extensions have previously been proposed to include such forces within existing ML frameworks^{66,67,68,69}, but for the purpose of this work, we use an explicitly shortrange model with an appropriately chosen cutoff.
The second advancement is the construction of suitable, physicallymotivated representations to predict the target properties of interest^{42,70,71,72,73}. In particular, the representation of an atomic configuration should reflect all the physical symmetries of the target property. The framework built around the Smooth Overlap of Atomic Positions (SOAP) descriptor^{74} and its covariant counterparts^{35,73}, which we call the atomcentered densitycorrelation framework, is well suited to the task of integrated machine learning modeling of multiple properties since it allows us to treat these properties within the same unified mathematical framework. We provide further details on the mathematical framework, as well as the construction of the unified ML model, including the definition of a polarizationderived collective variable, in the Methods section. Looking forward, the flexibility and extensibility of this framework will also allow us in future studies to include longrange interactions in a natural and general way, using a recent approach called LODE^{75,76}. This will allow us to address some of the observed disagreements with DFT benchmarks in the prediction of the phonon spectra, which likely derive from the neglect of longrange forces (see the subsection “Phonon dispersions” for additional details).
The modeling of multiple properties within a single ML framework is gaining importance as a way to extract richer information from simulations than the PES alone can provide. Such models combine the extensive, accurate, finitetemperature thermodynamic sampling afforded by an ML potential, as in a series of the previous works^{45,47,49,50,57,58,60,77,78}, with the expressiveness and utility of an ML property model. Particularly relevant are the studies using a potential energy surface combined with a dipolemoment model for studying the infrared spectra of isolated molecules^{79,80}. To date, such combined models have not yet been applied to ferroelectric materials; one important difficulty for ML modeling is the multivalued character of the polarization in the condensed phase (although see ref. ^{81} and ref. ^{82} for applications in liquid water, where this difficulty is much less severe). We describe a method of overcoming this difficulty in a systematic and generalizable way in the Methods section and in Supplementary Note 3. In the following study, we show how a combined modeling study can advance the field of ferroelectrics by providing a rich array of experimentally relevant properties from one unified mathematical framework.
Results
Summary
In this section, we summarize the main results obtained via the integrated ML model described in the Methods section. In the subsection “Structural transition in BaTiO_{3}”, we recover the wellknown sequence of ROTC phases and highlight the key role of the MLpredicted polarization vector in distinguishing each of the phases. In the subsection “The microscopic mechanism of the ferroelectric phase transition”, we find that the Ti offcentering is the driving mechanism of ferroelectricity and not just a result of cell distorsions.
In the subsection “Thermodynamics of BaTiO_{3}”, we provide an explicit calculation of the phase transition temperatures. Finally, in the subsection “Dielectric response of BaTiO_{3}”, we compute its temperature and frequencydependent dielectric properties and compare these with the experiment.
Structural transitions in BaTiO_{3}
The detection of the ferroelectric transitions of BaTiO_{3} in MD simulations is challenging due to the small lattice distortions and freeenergy differences that differentiate the phases. To overcome these challenges, one has to choose a sufficiently large cell so as to make the transitions clearly visible while allowing for a wellconverged statistical sampling of configurations across each of the coexistence regions. As a qualitative indicator of the phase transitions, we track the MD time evolution of the cell parameters (a, b, c, α, β, γ) and the histograms of the MLpredicted unitcell polarization components for each phase. The unitcell polarization correlates strongly with the magnitude and direction of the Tidisplacement (see SI).
Figure 1 shows that the hightemperature cubic phase consists of a collection of local minima arranged at the vertices of a cube, as proposed by the eightsite model^{5,13,83,84}. The presence of large thermal fluctuations, as compared to the energy barriers separating the minima, allows for diffusion of the polarization vector across the minima over a timescale of the order of a few ps in the MD trajectories. This makes the eight local Ti minima equally probable, yielding 〈P〉 = 0.
A reduction of the temperature results in a structural firstorder phase transition, both in agreement with ref. ^{26} and with previous experimental^{85} and theoretical works^{22,86} showing a divergence of the latent heat at the Curie point. Such transition is characterized by a clear breaking of reflection symmetry of the cell dipoles across one Cartesian (XY, YZ, or XZ) plane. The polarization vector can only visit four of the eight available cubic sites, marking the onset of the tetragonal phase. Any further decrease in the temperature further reduces the symmetry of the polarization histograms by successive freezing of the polarization components along a specific axis. At 50 K, the polarization densities show a single and broad minimum corresponding to an orthorhombic state, while at 15 K finally, the system is completely frozen in one minimum corresponding to the rhombohedral state. Note that each of the 6 tetragonal, 12 orthorhombic, and 8 rhombohedral minima that are trivially equivalent by symmetry can be reached depending on the initial configuration. These states are associated with different distortions of the lattice vectors (that are not symmetry invariants) and one can observe occasional transitions between them. Thus we can infer from Fig. 1 that the phenomenology of the ferroelectricparaelectric transition agrees well with the eightsite model. Under this model, the breakdown of ferroelectricity is characterized by thermal fluctuations across cubic offcentered sites that restore, on average, the centrosymmetry of the Tidisplacements.
Furthermore, we see that the action of a large isotropic pressure of 30 GPa in the cubic state fully restores the isotropy of the polarization densities (see Supplementary Fig. 4) and generates a paraelectric BaTiO_{3} phase down to 0 K. This is consistent with the experimentally observed loss of ferroelectricity in BaTiO_{3} at high pressures^{87}, as well as with the flattening of the calculated PES ^{15,16}, the disappearance of all unstable phonon modes of cubic BaTiO_{3}, and the isotropic Tidisplacement distribution observed in CarParrinello MD simulations of cubic BaTiO_{3} under pressure^{20}. This evidence strengthens the hypothesis that the fluctuations of the unitcell polarization between preferential orientations act as a microscopic precursor of the macroscopic ferroelectricity of the material.
Moreover, the emergence of a ferroelectric state is facilitated by the presence of spatial and directional correlations across the structures, which have been proposed to explain Xray diffraction results^{10,88} and directly observed in ref. ^{89}, with further evidence from firstprinciples calculations^{14,90,91}.
Figure 2 shows the extent and directionality of the spatial correlations (specifically, componentwise Pearson correlation coefficients; see Supplementary Note 5 for the exact expression) of the unitcell dipoles correlated against a central reference cell in the cubic phase at 250 K. The correlations are not only large and slowly decaying—they extend well up to the edges of the 5 × 5 × 5 supercell—but they are highly directional, with the slow decay taking place along the direction of the unitcell dipole.
It has long been assumed that these correlations arise from a combination of an Isinglike nearestneighbor interaction with a longrange dipole–dipole interaction, as typified e.g., in the model Hamiltonian of ref. ^{22}. Indeed, the authors of that study observed that the Coulomb interaction was critical for reproducing the ferroelectric ground state in their model—when it was turned off, the ground state became antiferroelectric. However, our simulations show longrange correlations and a ferroelectric ground state even though the energy model itself is explicitly shortranged—that is, the energy of an atom i is only sensitive to changes within a shortranged, local environment of 5.5 Å of the atom (see Methods, subsection “Training the ML model for BaTiO_{3}”)—making the correlations observed in our simulations an emergent phenomenon, not relying on the existence of any explicit longranged interaction. This range is sufficient to capture shortrange correlations between two neighboring Ti atoms, whose average distance in a typical MD run fluctuates about ≈4.0 Å. Furthermore, we observe these correlations even in the disordered, cubic phase, in contrast e.g., to ref. ^{91}, where the strongest correlations were observed only in the ordered phase (albeit in a different material, and where the phase transition was triggered not by temperature but by including quantum nuclear effects).
Thus, our understanding of the nature of ferroelectricity in BaTiO_{3} must take into account these emergent, longrange correlations. We will see, for instance, how they give rise to spontaneous ferroelectric states—even in the absence of lattice distortions—in the following subsection. On the other hand, these correlations also hamper the statistical and simulationcell size convergence of various quantities computed from statistical averages of the total dipole moment, as will be discussed in the subsection “Dielectric response of BaTiO_{3}”.
The microscopic mechanism of the ferroelectric transition
A fundamental question that arises in relation to the structural transitions observed in perovskite ferroelectrics concerns the driving mechanism of the transitions. We have seen in the previous subsection that the presence of the ferroelectric behavior is accompanied by the onset of a macroscopic polarization, mostly driven by ordered displacements of the transition metal atoms and cell deformation. This indicates that reducing the temperature makes it energetically favorable to develop polarized states, even at the expense of an internal strain introduced by the subsequent cell deformation.
One might, however, question whether it is the Ti offcentering or the cell deformation that drives this sequence of transitions or whether these distinct mechanisms are equally present and competing. To this end, ambientpressure simulations of a 4 × 4 × 4 cell over a wide range of temperatures (between 20 and 250 K) were carried out in a restricted cubic geometry. The geometric constraint inhibits the structural distortions, making it possible to investigate whether the displacements and ferroelectric states are still observable.
Twodimensional histograms of the Tidisplacements for a series of representative trajectories at 25, 80, 130, and 250 K are provided in Fig. 3. The lowesttemperature trajectory is equivalent to a fastfreezing experiment, where the Ti atoms relax to the closest potential energy minimum. We still observe offcentered states according to the eightsite model picture, but no transitions between neighboring cubic sites take place due to the negligible thermal fluctuations.
By slightly increasing the temperature, the thermal fluctuations are still smaller than the energy barrier between neighboring cubic sites but are sufficient to induce rare jumps between them. The Ti atoms consequently freeze in a local minimum, but notably, they all collectively jump to a single offcentered state after a small transient of the order of a few ps. The system stays trapped in this state for the whole simulation time.
This gives us evidence that the GAP inherently favors correlated ferroelectric displacements, despite the shortranged description of the interactions (in an Isinglike fashion) and indicates that a lowtemperature ferroelectric state arises in fully flexible simulations and in experiments as a consequence of a dipolar ordering, which is suppressed at hightemperatures by thermal fluctuations.
At higher temperatures, thermal fluctuations enable transitions between cubic sites with a rate that increases with the temperature. Due to the restored cubic centrosymmetry of the displacements, states with a net polarization are no longer observed, provided sufficiently long MD runs are performed. We find this same sequence of states in NVT cubic simulations (see Supplementary Note 7), showing that constraining the volume does not affect the qualitative picture of Fig. 3. Additionally, we note that the intermediate tetragonal and orthorhombic states do not occur in these simulations, as opposed to the fully flexible ones.
In conclusion, the presence of a dipolar ordering is responsible for the emergence of a lowtemperature ferroelectric state, as shown in Fig. 4 and in agreement with ref. ^{88}. At the same time, the absence of cell distortions considerably affects the shape of the FES, as the intermediate tetragonal and orthorhombic states do not occur with a fixed cell. This effect has also been reported in ref. ^{22}, where MonteCarlo simulation with no homogeneous strain showed the disappearance of such phases.
Thermodynamics of BaTiO_{3}
A challenge in modeling phase transitions such as the ones we focus on in this paper is that they are associated with small structural distortions that are comparable with the thermal fluctuations of individual atoms. A commonlyused strategy to improve the signaltonoise ratio is to use collective variables (CVs), such as the lattice parameters^{60}, which are naturally averaged over multiple atomic environments and directly reflect the macroscopic observables associated with the transition. Cell vectors, however, are not symmetryadapted so that multiple equivalent states are mapped to different values of the CVs. What is more, as we have seen in the previous subsection, cell distortions alone do not drive the different phase transitions of BaTiO_{3}, making them poor order parameters to distinguish these phases (see Supplementary Note 4 for a discussion of our metadynamics simulations that use a symmetrized combination of lattice parameters).
More effective characterization of the ferroelectric ordering can be obtained by explicitly using the predicted polarization P as an order parameter and, in particular, by building descriptors that show the orientation of the cell polarization relative to the atomic distortions. In Methods, subsection “Physicallyinspired order parameters”, we provide the construction of a twocomponent CV, namely s = (s_{1}, s_{2}), that gives us an effective lowdimensional description of the phases of BaTiO_{3}.
In Fig. 5, we show 2D contour lines of s across fully flexible MD runs of a BaTiO_{3} 4 × 4 × 4 supercell between 10 and 250 K. These represent molecular dynamics runs where the GAP predicts the coexistence of RO, OT, and TC states respectively, with comparable probability. Four distinct phases are clearly visible, showing how a polarizationderived twocomponent CV can easily identify the subtle differences between the four phases. The relative positions of the clusters give additional (but only qualitative) physical insights: as the CTO clusters are maximally distinguishable by s_{1} and the C center corresponds to the one with the lowest s_{1} value, the first CV is clearly related to the average polarization magnitude. This is predicted to be exactly zero for paraelectric cubic BaTiO_{3} in the thermodynamic limit while positive and increasingly large for the ferroelectric tetragonal and orthorhombic phases. The CV s_{1} can then be used to discriminate ferroelectric and paraelectric BaTiO_{3} states. Further evidence in this respect is provided in Supplementary Fig. 12, where we show how it is possible to reconstruct freeenergy profiles as a function of s_{1} across the TC transition and thus capture their finitetemperature stability.
On the other hand, s_{2} maximizes the difference between the ROT ferroelectric states; thus, we can relate it physically to the polarization orientation, in agreement with our observations of the subsection “Structural transitions in BaTiO_{3}”. Additional evidence for this interpretation is provided in Supplementary Fig. 8, where we analyze the correlation between the CVs, constructed here, with a meaningful physical observable, namely the polarization magnitude, in a series of 5 × 5 × 5 fully flexible trajectories.
Based on 2D maps as the one shown in Fig. 5, it is possible to cluster the MD trajectories and compute directly the temperaturedependent FES by calculating the relative concentration of the ROTC phases in sufficiently long MD runs so that many reversible transitions can be sampled. In practice, we never find more than two phases explored at each temperature. Fully flexible MD runs of a BaTiO_{3}4 × 4 × 4 cell, each with a total simulation time up to 1.6 ns, between 10 and 250 K, allow us to compute the relative Gibbs free energy \({{\Delta }}{G}^{k,k^{\prime} }(T)\) of the phase pair \((k,k^{\prime} )\) at temperature T and the corresponding chemical potential difference \({{\Delta }}{\mu }^{k,k^{\prime} }(T)\) using the following equations:
where N = 320 is the number of atoms, w(t) is the weight of the tth structure, P_{k}(t) is the probability that the tth structure belongs to the phase k, and k_{B} is the Boltzmann constant. For the OT and TC transition, unbiased MD simulations are used, i.e., with w(t) = 1, for every t. In the case of the RO transition, w(t) represents the weight of the tth structure computed via the iterative trajectory reweighting (ITRE) technique^{92}, which is used to remove the timedependent bias in the distribution of the microstates introduced by metadynamics.
The estimates of the critical temperatures at ambient pressure are computed by linear fits of the relative chemical potentials Δμ profiles, see Equation (1) and are reported for each pair of phases in Table 1. We note that our computed temperatures differ significantly from the experimentally observed transition temperatures, 393 K (TC), 278 K (OT), and 183 K (RO)^{2}. This underestimation of the critical temperatures, stemming from an underestimation of the freeenergy barriers between the phases, could come, in principle, from a variety of mechanisms, particularly the neglect of longrange electrostatics (and consequently of the LOTO splitting in the trainingset structures), as well as the presence of finitesize effects that could stabilize the hightemperature disordered phases in the MD. In fact, a significant size dependence of the finitetemperature properties of another perovskite, PbTiO_{3}, has been reported in a recent MLdriven study by ref. ^{93}.
However, previous work based on effective Hamiltonian models already pointed out this same underestimation of the critical temperatures and connected them to a shortcoming of the underlying exchangecorrelation functionals^{22,26}, which can be compensated by rescaling the potential energy surface or introducing an artificial negative pressure.
To confirm that pressure can significantly affect the transition temperature, in Fig. 9, we investigate the sensitivity of the Curie temperature to negative pressure (p = −2 GPa) within our ML framework. We observe a shift of the Curie temperature that is increased by a significant 82 K with respect to the ambient pressure estimate via NST simulations, while a very small variation in the lattice constant of the MD supercell (0.7% at 250 K) is seen in NpT. We note that the corresponding change in the volume (2.1%) is within the variation of calculated volumes of cubic BaTiO_{3} with different DFT exchangecorrelation functionals^{20}. Furthermore, experimental data^{94} on the elastic properties of BaTiO_{3} show that the bulk modulus of BaTiO_{3} is in the range of ≈200 GPa, implying that the action of relatively high pressure would result in a small change in the cell volumes while completely modifying the freeenergy landscape. This effect induces the Curie temperature shift. We note in this respect that the PBEsol functional that we used to compute energies and atomic forces of the trainingset structures, predicts a slightly underestimated lattice constant −4.0 Å at 400 K from CarParrinello MD^{20} as compared to the experimental value of 4.012 Å of cubic BaTiO_{3}^{95}. This is a minuscule underestimation that is, however, not negligible in the calculation of these tiny freeenergy barriers. Moreover, we also rule out that the main source of discrepancy with the experimental transition temperatures might arise from an incorrect prediction of thermal expansion (see Supplementary Fig. 11), as previously shown in ref. ^{23}, where the underestimation of the critical temperatures by an effective Hamiltonian model was related to the approximations made in the construction of the PES.
This sensitivity of the relative free energies on the equilibrium volume shows how it is possible to tune the applied pressure to obtain a better agreement with the experiment. Moreover, while the use of external pressure is a common strategy to improve the accuracy of ab initio MD—including in the recent MLdriven study by ref. ^{93}, to correct the socalled supertetragonality problem—our strategy opens up other avenues for improvement. For instance, since our ML potential is trained on a relatively small set of 1458 selfconsistent energy calculations, one could systematically test more accurate and demanding electronic structure approximations^{30,31,32,33}, by running them on the existing dataset to improve the quantitative agreement between simulations and experiment.
So far, we have shown the capability of the GAP of both qualitatively describing the emergence of ferroelectric states in BaTiO_{3} and reproducing the correct phase sequence. Seeing, however, the substantial disagreement between the critical temperature predictions of the ML model with the experiments, we shall now assess its accuracy as compared to the underlying DFT method. A compelling test in this direction is provided by the freeenergy perturbation (FEP) method. From the collected MD trajectories, we extract a validation set of 50 tetragonal and cubic structures just below (170 K) and above (194 K) the Curie point and recompute their energies with selfconsistent DFT calculations. This allows us to compute how the error of the GAPpredicted energies on the test set propagates to the error of the chemical potential estimate at a given temperature.
The FEP on the chemical potentials is first computed as a correction on the Gibbs free energy G^{k} of phase k = T, C:
where 〈⋅〉 represent the average over the test set structures, \({E}_{{{{\rm{GAP}}}}}^{k}{E}_{{{{\rm{DFT}}}}}^{k}\) the deviation between GAP and DFT total energies in phase k, and T the temperature. The FEP on the Gibbs free energies can then be translated into a correction on the chemical potential differences as follows:
where N is the number of atoms. Equation (2) represents an average of Boltzmann factors: if the energy deviations between the DFT and GAP estimates are small compared to the thermal fluctuations at temperature T for both the tetragonal and cubic phases, the correction on the corresponding chemical potential is negligible, due to the exponential factors. This propagation of errors can, however become significant or even dominant if the energy deviations are of the same order of magnitude or larger than the thermal fluctuations.
Panel (f) of Fig. 6 shows the effect of the FEP correction on the estimate of the chemical potentials for the two selected temperatures. The GAP shows good performance in the prediction of both tetragonal and cubic structures (compared to k_{B}T) and the FEP correction is one order of magnitude smaller than the actual prediction of Δμ and is still well within the error bars computed with the MD runs. The correction is hence negligible and no shift in the Curie point is observed, providing strong numerical evidence of the DFT accuracy of the GAP in freeenergy predictions.
Dielectric response of BaTiO_{3}
Let us now turn our attention to using the polarization model developed and described in Methods, subsection “Polarization model” to compute experimentally measurable quantities. As previously mentioned in ref. ^{96} and elsewhere in the literature on the modern theory of polarization^{97,98}, the polarization of a condensedphase system is well defined only modulo the quantum of polarization; however, we can still compute experimentally observable quantities as changes and fluctuations in its value.
The first of these experimentally relevant quantities is the static dielectric constant, which can be computed directly from the fluctuations of the system’s total dipole^{96}. In the cubic phase:
where both the total dipole magnitude M and the vacuum permittivity ε_{0} are expressed in SI units and the average value of the cell dipole \(\left\langle {{{\bf{M}}}}\right\rangle =0\) by symmetry. The optical (electronic) dielectric constant ε_{∞} from both measurements and calculations^{17,99,100} is in the range of 5 to 6, which is much smaller than the typical range of ε_{r} we calculate for this material, so this term will be neglected in the following analysis. In any case, the analyses below are nearly or completely insensitive to such a small constant shift. For noncubic phases, we must modify the equation to subtract off the (now nonzero) average polarization, replacing \(\left\langle {M}^{2}\right\rangle\) with \(\left\langle {M}_{\alpha }{M}_{\beta }\right\rangle \left\langle {M}_{\alpha }\right\rangle \left\langle {M}_{\beta }\right\rangle\), where α and β are Cartesian components of the total dipole vector^{101}. Notably, since the noncubic phases have an anisotropic structure, the dielectric tensor ε_{r,αβ} will also generally be anisotropic. Indeed, experimental measurements on singledomain crystals of BaTiO_{3} have shown a pronounced dielectric anisotropy, especially in the tetragonal phase^{2,102}.
Comparison of our results with experiments is complicated by the dramatic variation in the measured value with temperature, composition, and grain size^{103,104,105}. We, therefore, study the temperature dependence explicitly, as shown in Fig. 7. The calculated values for the orthorhombic, tetragonal, and cubic phases agree qualitatively with the calculations of ref. ^{100}, which used a similar computational methodology but with a shellmodel potential, as well as with measurements on singledomain crystals^{2,102}. In the tetragonal phase, we see the expected strong anisotropy between the components parallel (ε_{∥}) and perpendicular (ε_{⊥}) to the polarization axis—as we can already see from Fig. 1, the polarization fluctuations in the tetragonal phase are strongly suppressed along the polarization direction, which matches the much smaller value of ε_{∥} seen here. In the orthorhombic phase, the experimental measurements are averages over different domains and thus do not show the same pattern of anisotropy—namely, the splitting into three separate principal components—seen here, but this splitting is present in ref. ^{100}.
In the cubic phase, the expected temperature dependence follows a version of the Curie–Weiss law^{103}:
where T_{c,ε} is the (dielectric) Curie temperature, which should—in the limit of infinite system size and statistical sampling—agree with the tetragonalcubic phase transition temperature computed above, T_{c, C−T} = 182.4 K.
From the temperature dependence data in Fig. 7, we determine the bestfit parameters for the 4 × 4 × 4 cell data to be ε_{T→∞} = 90 ± 18, T_{c,ε} = (167.4 ± 2.3) K, and C = (57100 ± 3600) K. The most important discrepancy to note here is that the Curie point predicted by this fit is still about 15 K lower than the thermodynamic phase transition temperature predicted for ambient pressure in the subsection “Thermodynamics of BaTiO_{3}”. This discrepancy is likely a result of finitesize effects due to the small supercell, which are known to broaden and shift critical points^{106}. The 5 × 5 × 5 fit, on the other hand, yields ε_{T→∞} = 99 ± 18, T_{c,ε} = (171 ± 6) K, and C = (62200 ± 2500) K: the Curie temperature T_{c,ε} is slightly closer to the predicted phase transition temperature T_{c,C−T}, which is now within the 95% confidence interval of the fit parameters. However, even with the larger supercell, we still note a discrepancy from the parameters determined by fits to experimental data^{103,107}—namely, the Curie–Weiss constant C is underpredicted by a factor of about 2 with respect to the experiment. This difference could be due to approximations inherent in the underlying DFT functional, either directly or indirectly, due to the underestimation of the phase transition temperatures. We test this hypothesis in more detail below by investigating the negativepressure simulations.
The equation for the static dielectric constant, Equation (4), is, in fact, only the zerofrequency limit of the whole frequencydependent response function. We can compute the frequencydependent susceptibility (and thus the relative dielectric constant) via linear response theory from the onesided Fourier transform of the dipole–dipole autocorrelation function^{108,109} (again for the cubic phase):
where \({\tilde{C}}_{MM}(t)=\frac{1}{\left\langle {M}^{2}\right\rangle }\left\langle {{{\bf{M}}}}(0)\cdot {{{\bf{M}}}}(t)\right\rangle\) is the normalized dipole–dipole autocorrelation function and ε_{r} is the static dielectric constant computed from Equation (4).
We show the frequencydependent susceptibility for a 6 × 6 × 6 supercell trajectory at 250 K, computed using Equation (6), in Fig. 8. In general, we see the same structure as predicted for the hightemperature cubic phase by both theoretical effectiveHamiltonian MD calculations^{25} and observed experimentally^{103}, namely, that of a large absorption peak corresponding to the softmode (TO_{1}) phonon frequency. Note the slight negative dip in the real dielectric constant is expected and seen in many previous observations^{25,101,102}. This does not imply that the real or imaginary part of the refractive index \(n=\sqrt{{\epsilon }_{r}}\) is anywhere negative. It was previously proposed^{110,111} that the “softmode” part of the absorption spectrum of BaTiO_{3} could be described with a single, strongly damped harmonic oscillator of the form
with amplitude A, damping constant γ, and resonant frequency ω_{0} (which is always larger than the actual apparent peak frequency). However, a later study^{112} uncovered possible inadequacies of this singleoscillator model, especially in the highfrequency range (ω_{0} ≈ 100 cm^{−1}), and suggested a twooscillator model as a possible replacement, though it was not yet justified by the available experimental data.
More recently, ref. ^{25} both measured highaccuracy infrared spectra and computed theoretical spectra from MD simulations of the effective Hamiltonian model of ref. ^{28}, and they found strong evidence that the spectrum indeed is best modeled by two harmonic oscillators. The computed spectrum from our model at 250 K, shown in Fig. 8, further supports this picture: we also find that the imaginary part of the spectrum could only be satisfactorily described with two oscillators, although with different parameters from those calculated in ref. ^{25}: we find one oscillator with fundamental frequency ω_{1} = 86 cm^{−1} and damping ratio γ_{1}/ω_{1} = 3.0, and another with fundamental frequency ω_{2} = 187 cm^{−1} and damping ratio γ_{2}/ω_{2} = 1.1. Comparing these parameters with those calculated in ref. ^{25}, we find both frequencies to be rather high, so our agreement with their results remains mostly qualitative for now.
On the one hand, the differences we observe could be due to the large oscillations and lack of resolution at high frequencies due to the limited sampling time imposed by the relatively large computational cost of our model. However, it is more likely that both these discrepancies have the same origin as the underestimation of the phase transition temperatures discussed above – either inaccuracies in the underlying DFT model or some other effect not yet accounted for. As noted in the subsection “Thermodynamics of BaTiO_{3}”, the phase transition temperatures can be compensated by applying negative pressure. Indeed, ref. ^{25} associate the higherfrequency mode with shortrange correlations between (mostly) neighboring unitcell dipoles, so it is likely that this frequency shift has the same origin as the pressure effect.
To investigate this discrepancy further and to assess the effect of negative pressure on the dielectric response, we compute frequencydependent susceptibility spectra for all the negativepressure simulations previously run for subsection “Thermodynamics of BaTiO_{3}” (specifically, Fig. 9), where the material remained in the cubic phase. The spectra are also compared to those derived from ambientpressure simulations, specifically those used to compute the temperature dependence of the dielectric constant in Fig. 7. The comparison is shown in Fig. 10. On the one hand, we see the main peak shifting towards higher frequencies as the temperature increases, as expected from previous theoretical and experimental studies^{25,111}. On the other hand, we also see the peak shifting towards lower frequencies when negative pressure is applied at any given temperature. While the peak frequencies for the negativepressure simulations still do not match experimental data for the same temperatures, the shifts are in the right direction.
Furthermore, all simulations show a small narrow peak or edge at around 340 cm^{−1}, independent of the temperature. The frequency of the mode does depend on pressure, but due to the large bulk modulus of BaTiO_{3}, the mode shifts very little: only about 7 cm^{−1} under −2 GPa of pressure.
Although this peak likely represents a feature of our model and not just a simulation artifact, we do not yet have enough information to confidently identify this peak with known vibrational modes of BaTiO_{3}^{103,110,113}.
In fact, the difficulties we encounter here in reproducing the results of simpler, experimentally accurate—but empirically adjusted—models are reminiscent of the difficulties encountered previously, e.g., in ref. ^{68}, in applying more accurate (in the sense of reproducing the quantum PES) ML potentials that must, in turn, account for more accurate physics, such as manybody dispersion and quantum nuclear effects, in order to arrive at the right predictions for the right reasons. Rather than being a deficiency in the machine learning simulation approach, we see this as an opportunity to discover interesting physical behaviors and mechanisms that were overlooked before.
The calculations presented here are a promising first step towards using the MLPES and polarization framework as a generally applicable tool to predict experimentally relevant response properties. This tool will be a valuable future asset for investigating new candidate ferroelectric materials or gaining more insight into the underlying behavior of existing ones.
Discussion
In this work, we introduce a modern, general ML framework to describe at once the finitetemperature and functional properties (dielectric response) of perovskite ferroelectrics and apply it specifically to model barium titanate (BaTiO_{3}). This framework matches the accuracy of the underlying DFT method and does not require to preselect a given effective Hamiltonian model^{22,114}. The simulations made possible by this framework recover the correct ROTC phase ordering in fully flexible simulations and allow to investigate of the emergence of Ti offcenterings. In particular, we highlight the driving mechanism of the ferroelectric transition, showing how the presence of these offcentered displacements gives rise to a lowtemperature ferroelectric phase due to a longrange dipolar ordering. Moreover, the interplay between the displacements and the cell deformations leads to the emergence of intermediate tetragonal and orthorhombic phases. We further proceed to reconstruct the thermodynamics of BaTiO_{3} (see subsection “Thermodynamics of BaTiO_{3}”) by means of a twocomponent, polarizationderived CV.
Finally, we apply the ML polarization model to calculate dielectric response properties of experimental interest, including the static and frequencydependent dielectric constants, and investigate their dependence on temperature. While we do not reach a quantitative agreement with experimental measurements for many of the properties computed here, we see several clear, systematic pathways to improving the model potential and its predictions, such as including longrange electrostatic effects, simulating larger system sizes, as well as addressing the possible deficiencies in the underlying DFT model for both energies and polarizations. We expect that such improvements will allow us to reach a quantitative agreement with the experiments. Our results obtained with negative pressure calculations and the FEP show how this discrepancy can, in fact, be traced back in part to the sensitivity of the transition temperatures to cell volume combined with the deviation of the DFT cell volume from the experimental one. This effect suggests that a more indepth investigation of the effects of pressure—which is well known to influence the onset of ferroelectricity—could provide further insights into the deviation from experiments. A closer agreement could also be obtained by combining, as recently proposed, different DFT schemes to describe simultaneously the energy, structure, and electronic density of perovskite oxides^{115}.
We also plan to make improvements to the model performance by means of the feature sparsification technique, as detailed in ref. ^{70}. The latter has proven to reduce the computational cost (in energies and force predictions) by a factor of 3 or 4 for realistic systems and, in combination with largerscale parallelization techniques, will allow us to treat larger, more complex systems.
Importantly, this ML framework automates the construction of a model of the PES and the polarization and can then be used to investigate finitetemperature properties in detail and with firstprinciples accuracy. Since the MLPES was made with no explicit assumption on the functional form of the underlying PES and no prior definition of the relevant degrees of freedom of the system, this strategy is generalizable to other materials to study, e.g., 2D ferroelectrics^{29} and solid solutions with variable stoichiometries, that are known to possess different and more complex ferroelectric states. For instance, Ba_{x}Sr_{1−x}Ti_{y}Zr_{1−y}O_{3} is known to display a rich phase diagram, depending on composition, and shows both ferroelectric and relaxor ferroelectric phases^{116}. Furthermore, the framework developed is easily applicable to study the role of nuclear quantum effects, for instance, in incipient ferroelectrics such as SrTiO_{3} and KTaO_{3}, where quantum fluctuations appear to suppress the ferroelectric state^{91,117}. Further extensions of this framework include the investigation of the role of a finite electric field in the MD and its effect on polarization. This will allow us to simulate, for instance, hysteresis loops, which are key to measure the energy storage of ferroelectric devices.
In conclusion, we have shown how a comprehensive, datadriven modeling framework for a perovskite ferroelectric material, based on DFT reference data, can capture the mechanisms of the ferroelectric transition, as well as make predictions of thermodynamic and functional properties with firstprinciples accuracy. The work opens the door for a new avenue of fruitful research into the understanding and characterization of known ferroelectric materials, as well as the discovery and design of new candidate compounds with improved industrially relevant properties.
Methods
In this section, we first summarize the construction and properties of the symmetryadapted features used to train the ML models; a more thorough discussion of this family of features and an introduction to the notation we use here is given in Section 3 of ref. ^{73}. With these features defined, we detail how the potential energy surface and the polarization models are constructed. Turning our attention to the specifics of modeling BaTiO_{3}, we report the training and validation of our ML model. Furthermore, we develop physicallyinspired order parameters, which we use to characterize and interpret our results from subsection “Thermodynamics of BaTiO_{3}”. Finally, we report the computational details of the MLMD simulations.
Symmetryadapted features
To construct the family of features that are relevant for this paper, we make use of the atomcentered densitycorrelation framework^{118}. The starting point is the definition of a set of features, namely 〈anlm∣A; ρ_{i}〉, from an expansion of the atomic density for an environment i of structure A, as in Equation (31) of ref. ^{118}. The different indices in the bra identify the chemical species (a), radial function (n), and angular momentum (l, m), the latter being especially important to track the symmetry of the features.
Symmetryadapted descriptors can be obtained as a symmetrized average (referred to by an overline decoration) of the tensor product of ν sets of expansion coefficients, resulting in densitycorrelation features \(\langle q \overline{{\rho }_{i}^{\otimes \nu };\lambda \mu }\rangle\). While the generic index q only enumerates the features, the other indices encode the physical meaning of these descriptors. There are two fundamental parameters: (a) the bodyorder exponent ν, which indicates that the features describe the relative position of ν neighbors of the central atom and (b) the λ, μ coefficients, which determine how the descriptor transforms under rotations—namely as spherical harmonics \({Y}_{\lambda }^{\mu }\). This framework allows us to build features that are not only invariant to rotations but also explicitly covariant (more generally called equivariant) features of any tensor order. Such equivariant features were first introduced by ref. ^{119}, for vector features, and in ref. ^{35} for tensors of arbitrary order. Equivariant features are now gaining considerable popularity, especially for graph convolutional neural networks to predict scalar and tensor properties^{61,120,121,122,123}. In this work we only deal with spherical invariants or SOAP descriptors^{74}—corresponding to λ = 0, μ = 0 and λ = 1, μ = (−1, 0, +1) features, representing spherical equivariants of order 1. For instance, SOAP power spectrum features, which are invariant under rotations, are obtained from the contraction of two sets of coefficients (ν = 2):
These features can thus be written as \(\langle q A;\overline{{\rho }_{i}^{\otimes 2}}\rangle\).
Similarly, the simplest example of equivariant features only encodes information on the radial distribution of neighbors. They are equivalent to the density coefficients themselves:
An extension of this construction allows one to build symmetryadapted tensors of arbitrary rank and body order^{65}.
Given that, in order to learn dipole moments and polarizations, we only need the special case of vectorvalued features, we find it convenient to exploit the relationship between realvalued spherical harmonics of order λ = 1 and the Cartesian coordinates α = (x, y, z) to define Cartesian equivariants
The Cartesian equivariants of Equation (10) now explicitly transform as a 3vector under rotations:
\(\hat{R}A\) indicates an arbitrary rotation of a structure A, while \({R}_{\alpha \alpha ^{\prime} }\) is its representation as a 3 × 3 Cartesian matrix. We use this family of features to model the polarization of a BaTiO_{3} structure and to build an order parameter to distinguish the ROTC phases (see Results, subsection “Thermodynamics of BaTiO_{3}”). We refer the reader to refs. ^{70,73} and the documentation of librascal^{124} for implementation details.
Potential energy surface
A Gaussian approximation potential (GAP) is constructed by linear regression of energies E and atomic force components \({\left\{{{{{\bf{f}}}}}_{r}\right\}}_{r = 1}^{N}\), where N is the number of atoms, in the space of the kernels of these descriptors, representing the degree of correlation between the structures.
In order to control the computational cost of the calculation of energies and forces, we also construct a sparse set of representative atomic environments J that are used to define a basis of kernels k(⋅, J_{j}) in order to approximate the structureenergy relation. This is discussed further in subsection “Training the ML model for BaTiO_{3}”.
We write the target properties as a sum of kernel contributions:
where the kernel is built as a function of a set of atomcentered invariant features 〈q∣A_{i}〉, the index j runs over all environments J_{j} in the sparse set J and b_{j} are the weights on each sparse environment to be determined via ridge regression. Here we use SOAP power spectrum features, \(\langle q A;\overline{{\rho }_{i}^{\otimes 2}}\rangle\), and we compute the kernel between atomic environments as a scalar product raised to an integer power \(k({A}_{i},{A}_{i^{\prime} }^{\prime})={({\sum }_{q}\langle {A}_{i} q\rangle \langle q {A}_{i^{\prime} }^{\prime}\rangle )}^{\zeta }\), using ζ = 4 here, to introduce nonlinear behavior.
Polarization model
Besides this potential energy model, we construct a fully flexible, conformationally sensitive dipole moment surface for the material by employing the symmetryadapted Gaussian process regression (SAGPR) framework^{35}, previously benchmarked in the context of molecules in ref. ^{125} and proven to extend to the condensed phase in ref. ^{81}. Even though the cell polarization (or dipole) is not uniquely defined in periodic boundary conditions^{97,98}, we can still make a model for only a single branch of this polarization manifold with suitable preprocessing of the training data, detailed in Supplementary Note 3. This branch choice is essentially equivalent to fixing the polarization to be a single continuous function whose linearization about P = 0 is the product of Born effective charges and displacement from some nonpolar reference structure, in the spirit of ref. ^{126}.
As with existing SAGPR approaches, the total dipole of the cell is decomposed into vectorvalued atomic contributions. In analogy to Equation (12), we express the total dipole M and polarization P of a structure A as:
Our model works with total dipoles rather than polarizations as only the former are size extensive. A key advantage of this model is that we represent the dipole moment as a sum of atomcentered contributions (effectively, “partial dipoles”, in analogy to partial charges), giving us a spatially resolved, atomistic picture of how the different parts of the system contribute to the total polarization. Note that in contrast to the model described in ref. ^{125}, we do not define an additional partialcharge model, since such a model would depend on the choice of the unitcell and be incompatible with the modern theory of polarization. The only situation in which we use nonzero partial charges is in the linearized effectivecharge model used to shift the trainingset polarizations to the same branch; these effective charges are not used in the production model. As already remarked in ref. ^{96} and later in ref. ^{125}, this information can give us a much deeper insight into the physics of the system than predicting the total dipole alone. In this study, we use this information to define Ticentered unitcell dipoles by an appropriate sum of atomic partial dipoles. The dipole of the Tiatom is added to the dipoles from neighboring O and Ba atoms, with the neighboring contributions weighted (by 1/2 for O and 1/8 for Ba) so that the sum of the unitcell dipoles is still equal to the total cell dipole. These unitcell dipoles were used to make Figs. 1 and 2.
The model is trained on the same set of structures as the potential energy surface from subsection “Potential energy surface” but uses different training data and, generally, a different sparse set \(J^{\prime}\) each with a different set of weights {b}. These weights take the form of 3component vectors, corresponding to the kernel \(k({A}_{i},{A}_{i^{\prime} }^{\prime})\), which is now a rank2 Cartesian tensor (i.e., a 3 × 3 matrix) for any pair of environments. This kernel is computed, as in the scalar case, as an inner product of symmetryadapted features \(\langle q A;\overline{{\rho }_{i}^{\otimes 2};\alpha }\rangle\)
Training the ML model for BaTiO_{3}
As pointed out in the subsection “Potential energy surface”, constructing a GAP model requires defining a representative set of environments to control the computational cost in evaluating energies and atomic forces of structures. The representative environments should be ideally as diverse as possible so as to provide a good extrapolation across all the phases of interest. Specifically, for our case study of BaTiO_{3}, we use FarthestPoint Sampling (FPS) to select a total of 250 environments centered around barium and titanium atoms and 500 around oxygen atoms from the initial training dataset obtained via DFT optimizations (additional details are given below).
A second crucial parameter is the radial cutoff in the neighbor density ρ_{i}(x), defined in subsection “Symmetryadapted features”. This defines the size of the atomic environment, centered around atom i. Choosing large cutoff radii means including more neighbors in the density expansion and allows, in general, a more accurate representation of the environment. This happens, however, at the expense of increasing the computational complexity. For the purpose of constructing a GAP for BaTiO_{3}, we choose a radial cutoff of 5.5 Å around each center which is larger than the average separation of the first nearest Ti neighbors (≈4.0 Å). This cutoff allows us to capture the shortranged Ti–Ti interactions that ultimately result in longrange emergent dipole correlations, a distinctive feature of polarized states in BaTiO_{3}, as seen in Results, subsection “Structural transitions in BaTiO_{3}”.
The training dataset is constructed in an iterative fashion, which also means it can be systematically extended. Energies and forces are calculated using DFT as implemented in Quantum ESPRESSO^{127,128} with the PBEsol^{129} functional, and managed with AiiDA^{130,131,132}; further details can be found in Supplementary Note 2. An initial training set of N_{0} = 518 cubic structures (obtained from DFT optimizations with the PBEsol functional) is used to train a preliminary GAP. Molecular dynamics simulations with iPI^{133} are then performed in all the ROTC geometries and for a total simulation time of up to 500 ps. Among all uncorrelated structures thus generated with MD—the correlation being computed via the timedependent autocorrelation function of the total energy—only the most diverse according to their SOAP descriptors are then selected via FPS and recomputed with DFT selfconsistent calculations. These are then used to extend the training dataset and refit the GAP, thus restarting the loop and obtaining an increasingly accurate description of the PES. The final dataset built with this procedure has a total of 1458 structures, with an adequate sampling of all the phases of BaTiO_{3}. Specifically, on top of the initial training set of 518 structures, we added 100 structures coming from the first round of replicaexchange molecular dynamics (REMD) simulations in the NVT ensemble and additional 840 structures coming from a sampling of each of the ROTC phases (210 per phase) in the second round of NpT REMD calculations.
The learning curve of the GAP, trained on a total of 1200 training structures, is shown in Fig. 11a, with 258 randomly selected structures used as a validation set. The root mean square error (RMSE) decreases significantly with an increasing number of training points and the final accuracy of the potential in energy estimations is about 6 meV per formula unit (f.u.). This level of accuracy is sufficient to capture several interesting features of the physics of BaTiO_{3}, including the structural ROTC phases, the presence of needlelike correlations even in the hightemperature paraelectric phase, and to enable predictions of the freeenergy surface, that have the same degree of accuracy as the underlying DFT method (see Results, subsection “Thermodynamics of BaTiO_{3}”).
The polarization model, in contrast to the GAP, is trained only on the set of N_{T,pol} = 840 structures sampled from the NpT REMD calculations described above, with 210 structures coming from each of the four phases. A total of 200 randomly selected structures are withheld for testing; the largest model has therefore been trained with N_{max,pol} = 640 structures. The learning curve of the polarization model, shown in Fig. 11b, shows good performance; the largest model (N = 640 structures) achieves an accuracy of 3% of the intrinsic variation of the total dipoles in the training set, corresponding to an RMSE of 0.013 a_{0} per atom, or 0.07 a_{0} per unitcell—which is still small compared to the scale of unitcell polarizations shown, for example, in Fig. 1.
Phonon dispersions
A crucial test to evaluate the performance of the GAP is to compute phonon spectra and the corresponding density of states (DOS) and compare them with the DFT phonon spectra. In Fig. 12, we directly compare the outcome on a 4 × 4 × 4qmesh, taking two representative structures as reference: the 5atom cubic structure and the rhombohedral ground state, optimized via variablecell DFT calculations. The calculations were carried out via the finite difference method using the atomic simulation environment^{134} (ASE) for the GAP calculations and phonopy^{135} in conjunction with Quantum ESPRESSO for the DFT calculations. Since no explicit correction for the longrange electrostatics was explicitly taken into account in constructing the ML model, we compare the GAP predictions with the DFT calculations without such contributions. We stress, however, that this contribution due to longrange electrostatic interactions should be included to recover, e.g., the correct LO and TO mode splitting at Γ and to stabilize the TA mode of the rhombohedral structure along the TΓ and ΓF paths (see the Supplementary Fig. 10 for the DFT dispersion with LOTO splitting). It has been shown in the work of ref. ^{136} that shortranged potentials in polar materials can capture the correct phonon dispersions if the appropriate longrange dielectric model is subtracted before fitting the shortranged potential and then added back analytically—in analogy to what is done to Fourier interpolate phonon dispersions^{137}. We also show, in Supplementary Fig. 10, the full phonon spectra once these dielectric contributions are considered. The spectra show an overall good agreement, especially for the lowfrequency acoustic modes, with the most apparent discrepancies occurring for the highest LO mode. These discrepancies are likely to be caused by two main effects: (a) the training set construction and (b) the locality of the GAP. First, we recall that the interatomic potential is only trained on 2 × 2 × 2 structures so that longwavelength modes that correspond to the periodicity of a 4 × 4 × 4 cell lie in the extrapolative regime of the potential. Second, the GAP is only sensitive to atomic displacements within the chosen radial cutoff, so phonon modes with a small momentum q, and thus involving longwavelength excitations outside this radial cutoff, are not guaranteed to be well reproduced. These effects are likely the root of the disagreement between modes that lie along the ΓX and ΓL paths, like \((0,0,\frac{1}{4})\). Additional studies in this direction to investigate the role of the longrange electrostatic contribution on top of the GAP will shed light on this discrepancy and likely offer a better agreement with the reference DFT calculations. Furthermore, the inclusion of the LOTO splitting will allow us to perform a finitetemperature study of the phonon dispersion across the TC transition, to be compared with a recent study by ref. ^{138}. As we have seen, however, longrange electrostatic contributions are not essential to model the thermodynamics and phase transitions of BaTiO_{3}.
Validation with local dipole rotations
As a further test, we evaluate the accuracy of the GAP by modeling some of the distortions associated with the ferroelectric transition. In particular, the states associated with the presence of offcentered Ti atoms relative to the O cage and the energy barrier separating them is key. As we have discussed in Results, subsection “Structural transitions in BaTiO_{3}”, the longrange ordering of these displacements is the fundamental driver of ferroelectricity in BaTiO_{3}.
To test the performance of the GAP in reproducing these states, we construct two paths, representing a local dipole rotation, across the phase space of a 2 × 2 × 2 cubic supercell with a lattice parameter of 8 Å. We start with the DFToptimized structure with all Ti displaced by 0.082 Å along the 〈111〉 direction resulting in local dipoles, as depicted by the arrows in panel a of Fig. 13. This is a rhombohedral structure—spacegroup R3m (160)—with Ba and Ti occupying the 1a position (z_{Ba} = −0.0004 and z_{Ti} = 0.51116), and O occupying the 3b position (x_{O} = 0.48823, z_{O} = −0.01872). For reference, the cubic structure with no dipole moment would have z_{Ba} = 0, z_{Ti} = 0.5, x_{O} = 0.5, and z_{O} = 0, resulting in a cubic structure with spacegroup Pm\(\bar{3}\)m (221). One dipole, depicted in cyan, is then rotated about the barycenter of the enclosing oxygen octahedron to align with \(\langle \bar{1}\bar{1}\bar{1}\rangle\) while keeping the magnitude of the Tidisplacement constant and all other atoms fixed. The two paths, shown as the insets in panel b, have the same endpoints but visit different vertices of the cube centered at the barycenter of the octahedron with the \(\langle \bar{1}\bar{1}\bar{1}\rangle\) and 〈111〉 displacements defining a diagonal.
Physically, these paths represent the energy cost due to a relative rotation of one local dipole starting from a perfect ferroelectric state. A comparison between the GAP and the DFT energy variations across these paths (see panel b of Fig. 13) shows that the GAP correctly reproduces the energy profile and favors states that correspond to aligned Tidisplacements, a feature that we have also seen in lowtemperature MD simulations (see Results, subsection “The microscopic mechanism of the ferroelectric transition”). From a quantitative perspective, the GAP overestimates the energy barriers by some nonnegligible, but still reasonable, 18% for both paths. We stress, however, that these paths lie within the extrapolative regime of the potential, as they are constructed artificially and no MD simulation visits configurations that are close to them, except for the starting, completely ordered structure that is visited at low temperature (see Results, subsection “Structural transitions in BaTiO_{3}”).
Physicallyinspired order parameters
As mentioned in Results, subsection “Thermodynamics of BaTiO_{3}”, the construction of a CV that can effectively distinguish the structural phases of BaTiO_{3} is key for the prediction of its phase diagram. In this section, we provide the construction of a twocomponent CV, namely s = (s_{1}, s_{2}), by explicitly using the predicted polarization P as an order parameter. As we shall see, we will build a set of invariant descriptors that correspond, for each structure, to a scalar product of vectors. These are constructed using the equivariant features \(\langle an A;\overline{{\rho }_{i}^{\otimes \nu };\alpha }\rangle\) defined in Equation (10), averaged over Ticentered environments. Physically, they will carry information about the orientation of P relative to the ’mean’ atomic distortion, which we call Q (see also Fig. 14).
Firstly, in order to compute the CV efficiently for long MD runs, we need to define an easytocompute proxy for P, which we will denote as \(\tilde{{{{\bf{P}}}}}\). In practice, we find that some of the neighbor density coefficients 〈anlm∣A; ρ_{i}〉 introduced in the subsection “Symmetryadapted features” correlate strongly with P (see the correlation plots in Supplementary Fig. 7). We can then define \(\tilde{{{{\bf{P}}}}}\) by restricting ourselves to Ticentered environments, as follows:
where a = O represents the atomic species (the oxygen) onto which we project the Ticentered density. Note that here we use the expression for the Cartesian equivariants defined in subsection “Symmetryadapted features” so that α = (x, y, z) and \(\tilde{{{{\bf{P}}}}}\) transforms like a vector under rotations. It represents, in fact a sum of vectors \(\tilde{{P}_{i}}\), each assigned to one Ticenter, as shown in Fig. 14. Similarly, we average the full neighbor density coefficient \(\langle an \overline{{\rho }_{i}^{\otimes 1};\alpha }\rangle\) over all Ticenters to obtain a measure of the mean structural deformations:
Finally, we compute the scalar product of \(\tilde{{{{\bf{P}}}}}\) and \(\tilde{Q}\) to construct a set of invariants:
and perform a principal component analysis (PCA) on the scalar descriptors O^{an}(A) to obtain two physicallymotivated and symmetryinvariant order parameters. This step allows us to obtain the scalar components that mostly contribute to the observed variance of the O^{an} invariants across a dataset of structures. In particular, by performing a PCA analysis over the entirety of the MD trajectories as a function of all simulated temperatures, we find that the first two PCs, corresponding to s_{1} and s_{2} can neatly separate all four phases (see Results, subsection “Thermodynamics of BaTiO_{3}”).
At each temperature, we then perform a separate clustering using the probabilistic analysis of molecular motifs (PAMM)^{139} algorithm, that determines a Gaussian mixture model in which each cluster corresponds to a different phase. Using the posterior probabilities associated with the mixture model (named probabilistic motif identifiers in ref. ^{139}), we can associate with each MD frame a smooth probability P_{k}(t), based on the corresponding values of the CVs (s_{1}(t), s_{2}(t)), that represents the probability that the corresponding structure at time t belongs to the cluster k = (R, O, T, C). These probabilities are then used to determine the relative stability of the different phases. The advantage of this technique, as compared to perhaps simpler methodologies, such as tracking the temperature evolution of the lattice parameters, is the fact that it is fully automatized, rotationally invariant, and makes direct use of the polarization vector—the key ingredient to physically describe the onset of ferroelectricity.
MLMD Computational details
All the machine learning data that we have generated to investigate the physics of BaTiO_{3} combines the use of molecular dynamics simulations performed with iPI^{140}—the MD integrator—and librascal^{70,124}—the engine to compute the total energy, atomic force components, and stress tensor of a BaTiO_{3} structure. In all cases, we choose the smallest simulationcell size that provides converged results; this is to optimize the tradeoff between adequate sampling in time and adequate sampling in system size that is possible under a given computational budget.
In particular, the results of subsection “Structural transitions in BaTiO_{3}” correspond to NST fully flexible simulations of a 5 × 5 × 5 cell, i.e., with an external constant stress tensor σ = diag(p, p, p) with p = 1 atm. The full flexibility of the cell allows the system to relax the offdiagonal components of the MD computed stress tensor as the system undergoes the structural ROTC phase transitions as a function of the temperature. In this case, we choose the simulation size so as to show wellseparated structural minima as a function of the temperature while maintaining the simulations computationally inexpensive.
The results of Results, subsection “The microscopic mechanism of the ferroelectric transition“ correspond instead to isotropic NpT simulations of a 4 × 4 × 4 cell over a wide range of temperatures (between 20 and 250 K) with a restricted cubic geometry. This supercell size is sufficient to identify the Ti offcentering as the physical mechanism governing the emergence of ferroelectricity.
Fully flexible MD runs of a BaTiO_{3} 4 × 4 × 4 cell with a total simulation time up to 1.6 ns between 10 and 250 K are performed for quantitative estimation of the temperaturedependent free energies (see Results, subsection “Thermodynamics of BaTiO_{3}”). In particular, unbiased MD is used to generate trajectories across the coexistence regions of the OT and TC transitions (between 40 and 250 K), while welltempered metadynamics^{141} runs across the RO transition are needed to enable collective jumps between R and O states within times that are affordable by classical MD runs. Additional details on the metadynamics runs are given in Supplementary Note 4. The relatively small supercell size, in this case, allows both efficient sampling of the structural transitions and simulation times, on the order of nanoseconds, that are required to converge the chemical potential estimates.
The spatial correlations shown in Results, subsection “Structural transitions in BaTiO_{3}” are calculated on a 5 × 5 × 5 supercell trajectory of length 400 ps, while the static and frequencydependent dielectric constant in Results, subsection “Dielectric response of BaTiO_{3}” were calculated on a 6 × 6 × 6 supercell trajectory of length 200 ps in order to ensure supercellsize convergence of the static value. The temperature dependence of the dielectric constant, being a more expensive calculation requiring multiple trajectories, instead used both a 4 × 4 × 4 and a 5 × 5 × 5 supercell, simulated for 250 ps each, to explicitly assess the rate of supercellsize convergence.
All the NpT/NST simulations were carried out with an isotropic/anisotropic barostat, leaving the cell volume/vectors free to equilibrate at finite temperature. Thermalization of the cell degrees of freedom is achieved by means of a generalized Langevin equation (GLE) thermostat^{142}, while thermalization of the atomic velocity distribution is realized via stochastic velocity rescaling (SVR)^{143}. This combination of thermostats allows for an optimal equilibration of the system’s relevant degrees of freedom on a timescale of the order of picoseconds without significantly interfering with the dynamical properties of the system, especially the polarization vectors. The characteristic times of the barostat, the SVR thermostat, and the MD timestep are 1 ps, 10 fs, and 2 fs, respectively.
Data availability
All numerical data supporting the results of this paper and allowing to reproduce the results are openly available on the Materials Cloud Archive^{144}.
Code availability
In order to generate the data needed for this paper, we made use of the librascal^{124}, iPI^{133}, and TenSOAP^{145} codes. These codes are all publicly available on Github. Some additional scripts necessary for data analysis and processing beyond that provided in these codes is provided along with the research data in the Materials Cloud Archive^{144}.
References
Jona, F. & Shriane, G. Ferroelectric Crystals (Dover, 1962).
Merz, W. J. The electric and optical behavior of BaTiO_{3} singledomain crystals. Phys. Rev. 76, 1221–1225 (1949).
Cochran, W. Crystal stability and the theory of ferroelectricity. Adv. Phys. 9, 387–423 (1960).
Bersuker, I. B. PseudoJahnTeller effecta twostate paradigm in formation, deformation, and transformation of molecular systems and solids. Chem. Rev. 113, 1351–1390 (2013).
Bersuker, I. B. On the origin of ferroelectricity in perovskitetype crystals. Phys. Lett. 20, 589–590 (1966).
Chaves, A. S., Barreto, F. C. S., Nogueira, R. A. & Zéks, B. Thermodynamics of an eightsite orderdisorder model for ferroelectrics. Phys. Rev. B 13, 207–212 (1976).
Yamada, Y., Shirane, G. & Linz, A. Study of critical fluctuations in BaTiO_{3} by neutron scattering. Phys. Rev. 177, 848–857 (1969).
Vogt, H., Sanjurjo, J. A. & Rossbroich, G. Softmode spectroscopy in cubic BaTiO_{3} by hyperRaman scattering. Phys. Rev. B 26, 5904–5910 (1982).
Comès, R., Lambert, M. & Guinier, A. The chain structure of BaTiO_{3} and KNbO_{3}. Solid State Commun. 6, 715–719 (1968).
Comès, R., Lambert, M. & Guinier, A. Désordre linéaire dans les cristaux (cas du silicium, du quartz, et des pérovskites ferroélectriques). Acta Crystallogr. Sect. A 26, 244–254 (1970).
Paściak, M., Welberry, T., Kulda, J., Leoni, S. & Hlinka, J. Dynamic displacement disorder of cubic BaTiO_{3}. Phys. Rev. Lett. 120, 167601 (2018).
Girshberg, Y. & Yacoby, Y. Ferroelectric phase transitions and offcentre displacements in systems with strong electronphonon interaction. J. Phys. Condens. Matter 11, 9807–9822 (1999).
Pirc, R. & Blinc, R. Offcenter Ti model of barium titanate. Phys. Rev. B 70, 134107 (2004).
Paściak, M., Boulfelfel, S. E. & Leoni, S. Polarized cluster dynamics at the paraelectric to ferroelectric phase transition in BaTiO_{3}. J. Phys. Chem. B 114, 16465–16470 (2010).
Cohen, R. E. & Krakauer, H. Lattice dynamics and origin of ferroelectricity in BaTiO_{3}: linearizedaugmentedplanewave totalenergy calculations. Phys. Rev. B 42, 6416–6423 (1990).
Cohen, R. E. Origin of ferroelectricity in perovskite oxides. Nature 358, 136–138 (1992).
Ghosez, P., Gonze, X. & Michenaud, J.P. Lattice dynamics and ferroelectric instability of barium titanate. Ferroelectrics 194, 39–54 (1997).
Ghosez, P. H. S. H., Gonze, X. & Michenaud, J. P. Ab initio phonon dispersion curves and interatomic force constants of barium titanate. Ferroelectrics 206, 205–217 (1998).
Zhang, Q., Cagin, T. & Goddard, W. A. The ferroelectric and cubic phases in BaTiO_{3} ferroelectrics are also antiferroelectric. Proc. Natl Acad. Sci. USA 103, 14695–14700 (2006).
Kotiuga, M. et al. Microscopic picture of paraelectric perovskites from structural prototypes. Phys. Rev. Res. 4, L012042 (2022).
Zhao, X.G., Malyi, O. I., Billinge, S. J. L. & Zunger, A. Intrinsic local symmetry breaking in nominally cubic paraelectric BaTiO_{3}. Phys. Rev. B 105, 224108 (2022).
Zhong, W., Vanderbilt, D. & Rabe, K. M. Firstprinciples theory of ferroelectric phase transitions for perovskites: the case of BaTiO_{3}. Phys. Rev. B 52, 6301–6312 (1995).
Tinte, S., Íñiguez, J., Rabe, K. M. & Vanderbilt, D. Quantitative analysis of the firstprinciples effective Hamiltonian approach to ferroelectric perovskites. Phys. Rev. B 67, 064106 (2003).
Tinte, S., Stachiotti, M. G., Sepliarsky, M., Migoni, R. L. & Rodriguez, C. O. Atomistic modelling of BaTiO_{3} based on firstprinciples calculations. J. Phys. Condens. Matter 11, 9679–9690 (1999).
Ponomareva, I., Bellaiche, L., Ostapchuk, T., Hlinka, J. & Petzelt, J. Terahertz dielectric response of cubic BaTiO_{3}. Phys. Rev. B 77, 012102 (2008).
Qi, Y., Liu, S., Grinberg, I. & Rappe, A. M. Atomistic description for temperaturedriven phase transitions in BaTiO_{3}. Phys. Rev. B 94, 134308 (2016).
Krakauer, H., Yu, R., Wang, C.Z., Rabe, K. M. & Waghmare, U. V. Dynamic local distortions in KNbO_{3}. J. Phys. Condens. Matter 11, 3779–3787 (1999).
Walizer, L., Lisenkov, S. & Bellaiche, L. Finitetemperature properties of (Ba, Sr)TiO_{3} systems from atomistic simulations. Phys. Rev. B 73, 144105 (2006).
Zhang, J., Wei, D., Zhang, F., Chen, X. & Wang, D. Structural phase transition of two dimensional singlelayer SnTe from artificial neural network. Preprint at https://arxiv.org/abs/2012.11137 (2020).
Liechtenstein, A. I., Anisimov, V. I. & Zaanen, J. Densityfunctional theory and strong interactions: orbital ordering in MottHubbard insulators. Phys. Rev. B 52, 5467 (1995).
Dudarev, S. L., Botton, G. A., Savrasov, S. Y., Humphreys, C. J. & Sutton, A. P. Electronenergyloss spectra and the structural stability of nickel oxide: an LSDA+U study. Phys. Rev. B 57, 1505 (1998).
Perdew, J. P. Jacob’s ladder of density functional approximations for the exchangecorrelation energy. In AIP Conference Proceedings 1–20 (AIP, 2001).
Becke, A. D. A new mixing of hartreefock and local densityfunctional theories. J. Chem. Phys. 98, 1372 (1993).
Bartók, A. P., Payne, M. C., Kondor, R. & Csányi, G. Gaussian approximation potentials: the accuracy of quantum mechanics, without the electrons. Phys. Rev. Lett. 104, 136403 (2010).
Grisafi, A., Wilkins, D. M., Csányi, G. & Ceriotti, M. Symmetryadapted machine learning for tensorial properties of atomistic systems. Phys. Rev. Lett. 120, 036002 (2018).
Behler, J. & Parrinello, M. Generalized neuralnetwork representation of highdimensional potentialenergy surfaces. Phys. Rev. Lett. 98, 146401 (2007).
Rupp, M., Tkatchenko, A., Müller, K.R. & von Lilienfeld, O. A. Fast and accurate modeling of molecular atomization energies with machine learning. Phys. Rev. Lett. 108, 058301 (2012).
Montavon, G. et al. Machine learning of molecular electronic properties in chemical compound space. N. J. Phys. 15, 095003 (2013).
Behler, J. First principles neural network potentials for reactive simulations of large molecular and condensed systems. Angew. Chem. Int. Ed. 56, 12828–12840 (2017).
Deringer, V. L., Caro, M. A. & Csányi, G. Machine learning interatomic potentials as emerging tools for materials science. Adv. Mater. 31, 1902765 (2019).
Noé, F., Tkatchenko, A., Müller, K.R. & Clementi, C. Machine learning for molecular simulation. Annu. Rev. Phys. Chem. 71, 361–390 (2020).
Butler, K. T., Davies, D. W., Cartwright, H., Isayev, O. & Walsh, A. Machine learning for molecular and materials science. Nature 559, 547–555 (2018).
Schütt, K. T., Sauceda, H. E., Kindermans, P.J., Tkatchenko, A. & Müller, K.R. SchNet  A deep learning architecture for molecules and materials. J. Chem. Phys. 148, 241722 (2018).
Friederich, P., Häse, F., Proppe, J. & AspuruGuzik, A. Machinelearned potentials for nextgeneration matter simulations. Nat. Mater. 20, 750–761 (2021).
Lopanitsyna, N., Ben Mahmoud, C. & Ceriotti, M. Finitetemperature materials modeling from the quantum nuclei to the hot electron regime. Phys. Rev. Mater. 5, 043802 (2021).
Imbalzano, G. et al. Uncertainty estimation for molecular dynamics and sampling. J. Chem. Phys. 154, 074102 (2021).
Cheng, B., Engel, E. A., Behler, J., Dellago, C. & Ceriotti, M. Ab initio thermodynamics of liquid and solid water. Proc. Natl Acad. Sci. USA 116, 1110–1115 (2019).
Carleo, G. et al. Machine learning and the physical sciences. Rev. Mod. Phys. 91, 045002 (2019).
Dragoni, D., Daff, T. D., Csányi, G. & Marzari, N. Achieving DFT accuracy with a machinelearning interatomic potential: thermomechanics and defects in bcc ferromagnetic iron. Phys. Rev. Mater. 2, 013808 (2018).
Bartók, A. P., Kermode, J., Bernstein, N. & Csányi, G. Machine learning a generalpurpose interatomic potential for silicon. Phys. Rev. X 8, 041048 (2018).
Isayev, O. et al. Materials cartography: representing and mining materials space using structural and electronic fingerprints. Chem. Mater. 27, 735–743 (2015).
SanchezLengeling, B. & AspuruGuzik, A. Inverse molecular design using machine learning: generative models for matter engineering. Science 361, 360–365 (2018).
Szlachta, W. J., Bartók, A. P. & Csányi, G. Accuracy and transferability of Gaussian approximation potential models for tungsten. Phys. Rev. B 90, 104108 (2014).
Deringer, V. L. & Csányi, G. Machine learning based interatomic potential for amorphous carbon. Phys. Rev. B 95, 094203 (2017).
Morawietz, T., Singraber, A., Dellago, C. & Behler, J. How van der waals interactions determine the unique properties of water. Proc. Natl Acad. Sci. USA 113, 8368–8373 (2016).
Caro, M. A., Deringer, V. L., Koskinen, J., Laurila, T. & Csányi, G. Growth mechanism and origin of high sp^{3} content in tetrahedral amorphous carbon. Phys. Rev. Lett. 120, 166101 (2018).
Sosso, G. C., Miceli, G., Caravati, S., Behler, J. & Bernasconi, M. Neural network interatomic potential for the phase change material GeTe. Phys. Rev. B 85, 174103 (2012).
Eshet, H., Khaliullin, R. Z., Kühne, T. D., Behler, J. & Parrinello, M. Microscopic origins of the anomalous melting behavior of sodium under high pressure. Phys. Rev. Lett. 108, 115701 (2012).
Khaliullin, R. Z., Eshet, H., Kühne, T. D., Behler, J. & Parrinello, M. Graphitediamond phase coexistence study employing a neuralnetwork mapping of the ab initio potential energy surface. Phys. Rev. B 81, 100103 (2010).
Jinnouchi, R., Lahnsteiner, J., Karsai, F., Kresse, G. & Bokdam, M. Phase transitions of hybrid perovskites simulated by machinelearning force fields trained on the fly with Bayesian inference. Phys. Rev. Lett. 122, 225701 (2019).
Park, C. W. et al. Accurate and scalable graph neural network force field and molecular dynamics with direct force architecture. npj Comput. Mater. 7, 1–9 (2021).
Thompson, A., Swiler, L., Trott, C., Foiles, S. & Tucker, G. Spectral neighbor analysis method for automated generation of quantumaccurate interatomic potentials. J. Comput. Phys. 285, 316–330 (2015).
Shapeev, A. V. Moment tensor potentials: a class of systematically improvable interatomic potentials. Multiscale Model. Simul. 14, 1153–1173 (2016).
van der Oord, C., Dusson, G., Csányi, G. & Ortner, C. Regularised atomic bodyordered permutationinvariant polynomials for the construction of interatomic potentials. Mach. Learn. Sci. Technol. 1, 015004 (2020).
Nigam, J., Pozdnyakov, S. & Ceriotti, M. Recursive evaluation and iterative contraction of N body equivariant features. J. Chem. Phys. 153, 121101 (2020).
Artrith, N., Morawietz, T. & Behler, J. Highdimensional neuralnetwork potentials for multicomponent systems: applications to zinc oxide. Phys. Rev. B 83, 153101 (2011).
Bereau, T., DiStasio, R. A., Tkatchenko, A. & von Lilienfeld, O. A. Noncovalent interactions across organic and biological subsets of chemical space: physicsbased potentials parametrized from machine learning. J. Chem. Phys. 148, 241706 (2018).
Veit, M. et al. Equation of state of fluid methane from first principles with machine learning potentials. J. Chem. Theory Comput. 15, 2574–2586 (2019).
Ko, T. W., Finkler, J. A., Goedecker, S. & Behler, J. A fourthgeneration highdimensional neural network potential with accurate electrostatics including nonlocal charge transfer. Nat. Commun. 12, 398 (2021).
Musil, F. et al. Efficient implementation of atomdensity representations. J. Chem. Phys. 154, 114109 (2021).
Himanen, L., Geurts, A., Foster, A. S. & Rinke, P. Datadriven materials science: status, challenges, and perspectives. Adv. Sci. 6, 1900808 (2019).
Csányi, G., Willatt, M. J. & Ceriotti, M. in Machine Learning Meets Quantum Physics (eds Schütt, K. T. et al.) Ch. 6 (Springer International Publishing, 2020).
Musil, F. et al. Physicsinspired structural representations for molecules and materials. Chem. Rev. 121, 9759–9815 (2021).
Bartók, A. P., Kondor, R. & Csányi, G. On representing chemical environments. Phys. Rev. B 87, 184115 (2013).
Grisafi, A. & Ceriotti, M. Incorporating longrange physics in atomicscale machine learning. J. Chem. Phys. 151, 204105 (2019).
Grisafi, A., Nigam, J. & Ceriotti, M. Multiscale approach for the prediction of atomic scale properties. Chem. Sci. 12, 2078–2090 (2021).
Imbalzano, G. & Ceriotti, M. Modeling the Ga/As binary system across temperatures and compositions from first principles. Phys. Rev. Mater. 5, 063804 (2021).
Deringer, V. L. et al. Origins of structural and electronic transitions in disordered silicon. Nature 589, 59–64 (2021).
Gastegger, M., Behler, J. & Marquetand, P. Machine learning molecular dynamics for the simulation of infrared spectra. Chem. Sci. 8, 6924–6935 (2017).
Laurens, G., Rabary, M., Lam, J., Peláez, D. & Allouche, A.R. Infrared spectra of neutral polycyclic aromatic hydrocarbons based on machine learning potential energy surface and dipole mapping. Theor. Chem. Acc. 140, 66 (2021).
Kapil, V., Wilkins, D. M., Lan, J. & Ceriotti, M. Inexpensive modeling of quantum dynamics using path integral generalized Langevin equation thermostats. J. Chem. Phys. 152, 124104 (2020).
Zhang, L. et al. Deep neural network for the dielectric response of insulators. Phys. Rev. B 102, 041121 (2020).
Chaves, A. S., Barreto, F. C. S., Nogueira, R. A. & Zẽks, B. Thermodynamics of an eightsite orderdisorder model for ferroelectrics. Phys. Rev. B 13, 207–212 (1976).
Comes, R., Lambert, M. & Guinier, A. The chain structure of BaTiO_{3} and KNbO_{3}. Solid State Commun. 6, 715–719 (1968).
Roberts, S. Adiabatic study of the 128° C transition in barium titanate. Phys. Rev. 85, 925–926 (1952).
Zhong, W., Vanderbilt, D. & Rabe, K. M. Phase transitions in BaTiO_{3} from first principles. Phys. Rev. Lett. 73, 1861–1864 (1994).
Decker, D. L. & Zhao, Y. X. Dielectric and polarization measurements on BaTiO_{3} at high pressures to the tricritical point. Phys. Rev. B 39, 2432–2438 (1989).
Senn, M., Keen, D., Lucas, T., Hriljac, J. & Goodwin, A. Emergence of longrange order in batio_{3} from local symmetrybreaking distortions. Phys. Rev. Lett. 116, 207602 (2016).
Bencan, A. et al. Atomic scale symmetry and polar nanoclusters in the paraelectric phase of ferroelectric materials. Nat. Commun. 12, 3509 (2021).
Vanderbilt, D. & Zhong, W. Firstprinciples theory of structural phase transitions for perovskites: competing instabilities. Ferroelectrics 206, 181–204 (1998).
Akbarzadeh, A. R., Bellaiche, L., Leung, K., Íñiguez, J. & Vanderbilt, D. Atomistic simulations of the incipient ferroelectric KTaO_{3}. Phys. Rev. B 70, 054103 (2004).
Giberti, F., Cheng, B., Tribello, G. A. & Ceriotti, M. Iterative unbiasing of quasiequilibrium sampling. J. Chem. Theory Comput. 16, 100–107 (2020).
Xie, P., Chen, Y., E, W. & Car, R. Ab initio multiscale modeling of ferroelectrics: the case of PbTiO_{3} Preprint at https://arxiv.org/abs/2205.11839 (2022).
Fischer, G. J., Wang, Z. & Karato, S.i Elasticity of CaTiO_{3}, SrTiO_{3} and BaTiO_{3} perovskites up to 3.0 Gpa: the effect of crystallographic structure. Phys. Chem. Miner. 20, 97–103 (1993).
Kay, H. & Vousden, P. XCV. Symmetry changes in barium titanate at low temperatures and their relation to its ferroelectric properties. Lond. Edinb. Dublin Philos. Mag. J. Sci. 40, 1019–1040 (1949).
Sharma, M., Resta, R. & Car, R. Dipolar correlations and the dielectric permittivity of water. Phys. Rev. Lett. 98, 247401 (2007).
Resta, R. & Vanderbilt, D. In Physics of Ferroelectrics: A Modern Perspective (eds Rabe, K. M., Ahn, C. H. & Triscone, J.M.) pp. 31–68 (Springer, 2007).
Spaldin, N. A. A beginner’s guide to the modern theory of polarization. J. Solid State Chem. 195, 2–10 (2012).
Zhong, W., Vanderbilt, D. & Rabe, K. M. Firstprinciples theory of ferroelectric phase transitions for perovskites: the case of BaTiO_{3}. Phys. Rev. B 52, 6301–6312 (1995).
Hashimoto, T. & Moriwake, H. Dielectric properties of BaTiO_{3} by molecular dynamics simulations using a shell model. Mol. Simul. 41, 1074–1080 (2015).
MacDowell, L. G. & Vega, C. Dielectric constant of ice Ih and ice V: a computer simulation study. J. Phys. Chem. B 114, 6089–6098 (2010).
Li, Z., Grimsditch, M., Foster, C. M. & Chan, S. K. Dielectric and elastic properties of ferroelectric materials at elevated temperature. J. Phys. Chem. Solids 57, 1433–1438 (1996).
Ostapchuk, T., Petzelt, J., Savinov, M., Buscaglia, V. & Mitoseriu, L. Grainsize effect in BaTiO_{3} ceramics: study by far infrared spectroscopy. Phase Transit. 79, 361–373 (2006).
Davis, L. & Rubin, L. G. Some dielectric properties of bariumstrontium titanate ceramics at 3000 megacycles. J. Appl. Phys. 24, 1194–1197 (1953).
Chu, F., Sun, H.T., Zhang, L.Y. & Yao, X. Temperature dependence of ultralowfrequency dielectric relaxation of barium titanate ceramic. J. Am. Ceram. Soc. 75, 2939–2944 (1992).
Binder, K. Finite size effects on phase transitions. Ferroelectrics 73, 43–67 (1987).
Rupprecht, G. & Bell, R. O. Dielectric constant in paraelectric perovskites. Phys. Rev. 135, A748–A752 (1964).
Löffler, G., Schreiber, H. & Steinhauser, O. The frequencydependent conductivity of a saturated solution of ZnBr_{2} in water: a molecular dynamics simulation. J. Chem. Phys. 107, 3135–3143 (1997).
Frenkel, D. Understanding Molecular Simulation: From Algorithms to Applications 2nd edn (Academic Press, 2002).
Luspin, Y., Servoin, J. L. & Gervais, F. Soft mode spectroscopy in barium titanate. J. Phys. C Solid State Phys. 13, 3761–3773 (1980).
Vogt, H., Sanjurjo, J. A. & Rossbroich, G. Softmode spectroscopy in cubic BaTiO_{3} by hyperraman scattering. Phys. Rev. B 26, 5904–5910 (1982).
Presting, H., Sanjurjo, J. A. & Vogt, H. Mode softening in cubic BaTiO_{3} and the problem of its adequate description. Phys. Rev. B 28, 6097–6099 (1983).
Hlinka, J., Petzelt, J., Kamba, S., Noujni, D. & Ostapchuk, T. Infrared dielectric response of relaxor ferroelectrics. Phase Transit. 79, 41–78 (2006).
García, A. & Vanderbilt, D. Temperaturedependent dielectric response of BaTiO_{3} from first principles. AIP Conf. Proc. 436, 53–60 (1998).
Williams, K., Wagner, L. K., Cazorla, C. & Gould, T. Combining density functional theories to correctly describe the energy, lattice structure and electronic density of functional oxide perovskites. Preprint at https://arxiv.org/abs/2005.03792 (2020).
Maiti, T., Guo, R. & Bhalla, A. S. Structureproperty phase diagram of BaZr_{x}Ti_{1−x}O_{3} system. J. Am. Ceram. Soc. 91, 1769–1780 (2008).
Zhong, W. & Vanderbilt, D. Effect of quantum fluctuations on structural phase transitions in SrTiO_{3} and BaTiO_{3}. Phys. Rev. B Condens. Matter Mater. Phys. 53, 5047–5050 (1996).
Willatt, M. J., Musil, F. & Ceriotti, M. Atomdensity representations for machine learning. J. Chem. Phys. 150, 154110 (2019).
Glielmo, A., Sollich, P. & De Vita, A. Accurate interatomic force fields via machine learning with covariant kernels. Phys. Rev. B 95, 214302 (2017).
Anderson, B., Hy, T. S. & Kondor, R. Cormorant: covariant molecular neural networks. In 33rd Conference on Neural Information Processing Systems (eds Wallach, H. et al.) (Curran Associates, Inc., 2019).
Batzner, S. et al. E(3)equivariant graph neural networks for dataefficient and accurate interatomic potentials. Nat. Commun. 13, 2453 (2022).
Schütt, K., Unke, O. & Gastegger, M. Equivariant message passing for the prediction of tensorial properties and molecular spectra. in Proc. 38th International Conference on Machine Learning (eds Meila, M. & Zhang, T.) 9377–9388 (PMLR, 2021).
Qiao, Z. et al. Informing geometric deep learning with electronic interactions to accelerate quantum chemistry. Proc. Natl Acad. Sci. USA 119, e2205221119 (2022).
Musil, F. et al. librascal. https://github.com/cosmoepfl/librascal (2020).
Veit, M., Wilkins, D. M., Yang, Y., DiStasio, R. A. & Ceriotti, M. Predicting molecular dipole moments by combining atomic partial charges and atomic dipoles. J. Chem. Phys. 153, 024113 (2020).
Zhong, W., KingSmith, R. D. & Vanderbilt, D. Giant LOTO splittings in perovskite ferroelectrics. Phys. Rev. Lett. 72, 3618–3621 (1994).
Giannozzi, P. et al. Quantum espresso: a modular and opensource software project for quantum simulations of materials. J. Phys. Condens. Matter 21, 395502 (2009).
Giannozzi, P. et al. Advanced capabilities for materials modelling with quantum espresso. J. Phys. Condens. Matter 29, 465901 (2017).
Perdew, J. P. et al. Restoring the densitygradient expansion for exchange in solids and surfaces. Phys. Rev. Lett. 100, 136406 (2008).
Pizzi, G., Cepellotti, A., Sabatini, R., Marzari, N. & Kozinsky, B. Aiida: automated interactive infrastructure and database for computational science. Comput. Mater. Sci. 111, 218–230 (2016).
Huber, S. P. et al. Aiida 1.0, a scalable computational infrastructure for automated reproducible workflows and data provenance. Sci. Data 7, 300 (2020).
Uhrin, M., Huber, S. P., Yu, J., Marzari, N. & Pizzi, G. Workflows in aiida: engineering a highthroughput, eventbased engine for robust and modular computational workflows. Comput. Mater. Sci. 187, 110086 (2021).
Kapil, V. et al. IPI Software. http://ipicode.org (2018).
Larsen, A. H. et al. The atomic simulation environmenta Python library for working with atoms. J. Phys. Condens. Matter 29, 273002 (2017).
Togo, A. & Tanaka, I. First principles phonon calculations in materials science. Scr. Mater. 108, 1–5 (2015).
Libbi, F., Bonini, N. & Marzari, N. Thermomechanical properties of honeycomb lattices from internalcoordinates potentials: the case of graphene and hexagonal boron nitride. 2D Mater. 8, 015026 (2020).
Giannozzi, P., de Gironcoli, S., Pavone, P. & Baroni, S. Ab initio calculation of phonon dispersions in semiconductors. Phys. Rev. B 43, 7231–7242 (1991).
Zhang, X., Zhang, C., Zhang, C., Zhang, P. & Kang, W. Finitetemperature phonon dispersion and vibrational dynamics of BaTiO_{3} from firstprinciples molecular dynamics. Phys. Rev. B 105, 014304 (2022).
Gasparotto, P., Meißner, R. H. & Ceriotti, M. Recognizing local and global structural motifs at the atomic scale. J. Chem. Theory Comput. 14, 486–498 (2018).
Kapil, V. et al. IPI 2.0: a universal force engine for advanced molecular simulations. Comput. Phys. Commun. 236, 214–223 (2019).
Barducci, A., Bussi, G. & Parrinello, M. Welltempered metadynamics: a smoothly converging and tunable freeenergy method. Phys. Rev. Lett. 100, 020603 (2008).
Ceriotti, M., Manolopoulos, D. E. & Parrinello, M. Accelerating the convergence of path integral dynamics with a generalized Langevin equation. J. Chem. Phys. 134, 84104 (2011).
Bussi, G., Donadio, D. & Parrinello, M. Canonical sampling through velocity rescaling. J. Chem. Phys. 126, 14101 (2007).
Gigli, L. et al. Thermodynamics and dielectric response of BaTiO_{3} by datadriven modeling. Materials Cloud Archive 2022.88, https://doi.org/10.24435/materialscloud:9gk6 (2022).
Wilkins, D. M. & Grisafi, A. TENSOAP repository. https://github.com/dilkins/TENSOAP (2021).
Hinuma, Y., Pizzi, G., Kumagai, Y., Oba, F. & Tanaka, I. Band structure diagram paths based on crystallography. Comput. Mater. Sci. 128, 140–184 (2017).
Acknowledgements
We thank Federico Grasselli for his insightful suggestions and critical reading of the manuscript. L.G., M.K. and M.C. were supported by the Samsung Advanced Institute of Technology (SAIT). M.V., G.P., N.M. and M.C. acknowledge support from the MARVEL National Centre of Competence in Research (NCCR), funded by the Swiss National Science Foundation (grant agreement ID 51NF40182892). G.P. acknowledges the swissuniversities “Materials Cloud” project (number 201003). G.P. and N.M. acknowledge support from the European Centre of Excellence MaX “Materials design at the Exascale” (824143). This work was supported by a grant from the Swiss National Supercomputing Centre (CSCS) under project IDs mr0 and s1073.
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G.P., N.M. and M.C. jointly supervised the project. L.G. and M.V. jointly developed, trained, and benchmarked the ML framework. L.G. ran the MD simulations. L.G. and M.V. analyzed the results of the MD simulations. M.K. performed the DFT calculations. All authors contributed to the discussion and writing of the paper.
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Gigli, L., Veit, M., Kotiuga, M. et al. Thermodynamics and dielectric response of BaTiO_{3} by datadriven modeling. npj Comput Mater 8, 209 (2022). https://doi.org/10.1038/s41524022008450
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DOI: https://doi.org/10.1038/s41524022008450
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